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9.2 Variation of the transconductance with the interface state density

Close to the threshold voltage the linear transconductance reaches a maximum when its increase in the subthreshold region ( $\left |V_\mathrm {GS}\right | < \left |V_\mathrm {TH}\right |$ ) is compensated by its decrease in the triode region ( $\left |V_\mathrm {GS}\right | \geq \left |V_\mathrm {TH}\right |$ ) of the device. This maximum value of the transconductance may be approximated in the linear regime as follows:

$\{begin}{align*} g_\mathrm {m,lin}^\mathrm {max} &\approx -\frac {W}{L} \frac {\mu _\mathrm {0} C_\mathrm {OX} V_\mathrm {DS}}{1+\alpha _0 N_\mathrm {it}}. \{end}{align*}$

The variation of the maximum transconductance with the increase in interface state density may be calculated by differentiating the maximum transconductance:

$\{begin}{align*} \frac {\partial {g_\mathrm {m,lin}^\mathrm {max}}}{\partial {N_\mathrm {it}}} &\approx \frac {W}{L} \mu _\mathrm {0} C_\mathrm {OX} V_\mathrm {DS} \frac {\alpha _0}{(1+\alpha _0 N_\mathrm {it})^2}, \\ \frac {\Delta {g_\mathrm {m,lin}^\mathrm {max}}}{g_\mathrm {m,lin}^\mathrm {max}} &\approx -\frac {\alpha _0 \Delta {N_\mathrm {it}}}{1+\alpha _0 N_\mathrm {it}}. \{end}{align*}$

In the saturation regime the transconductance does not reach a maximum which makes it considerably difficult to evaluate an appropriate operating point to refer to. The absence of this reference point is the reason why mobility degradation is usually analyzed in the linear drain current regime and not in the saturation regime. Close to the threshold voltage the saturation transconductance may be approximated as follows:

$\{begin}{align*} g_\mathrm {m,sat}^\mathrm {V_\mathrm {TH}} &\approx -2 \frac {W}{L} \frac {\mu _\mathrm {0} C_\mathrm {OX} (V_\mathrm {GS}-V_\mathrm {TH})}{1+\alpha _0 N_\mathrm {it}}. \{end}{align*}$

The variation of this saturation transconductance with the increase in interface state density may be calculated by differentiating the upper equation:

$\{begin}{align*} \frac {\partial {g_\mathrm {m,sat}^\mathrm {V_\mathrm {TH}}}}{\partial {N_\mathrm {it}}} &\approx 2 \frac {W}{L} \frac {\mu _\mathrm {0} C_\mathrm {OX} (V_\mathrm {GS}-V_\mathrm {TH})}{1+\alpha _0 N_\mathrm {it}} \left (\frac {\alpha _0}{1+\alpha _0 N_\mathrm {it}} + \frac {1}{V_\mathrm {GS}-V_\mathrm {TH}} \frac {\partial {V_\mathrm {TH}}}{\partial {N_\mathrm {it}}}\right ), \\ \frac {\Delta {g_\mathrm {m,sat}^\mathrm {V_\mathrm {TH}}}}{g_\mathrm {m,sat}^\mathrm {V_\mathrm {TH}}} &\approx -\frac {\alpha _0 \Delta {N_\mathrm {it}}}{1+\alpha _0 N_\mathrm {it}} - \frac {\Delta {V_\mathrm {TH}^q}}{V_\mathrm {GS}-V_\mathrm {TH}}. \{end}{align*}$

Note that both equations are very crude approximations which are only valid close to the threshold voltage of the device.

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